THIS IS AN OLD VERSION OF THE PAGE (OCT. 2001 - JULY 2002.)
*** WELCOME TO FOURIER SERIES & INTEGRAL TRANSFORMS ***
*** ( F. S. I. T. ) ***
(The page for earlier semesters, (i.e. for October 2001 and earlier)
for Fourier S. I. T. is
Some information about the examination on July 12, 2002 is
We wish you a very pleasant, interesting and successful semester. We
realize that the strike will make this considerably harder to achieve.
Within the constraints of the situation, we will do whatever we can to
minimize the difficulties, and also to help you know what you need for
the continuation of your studies next semester.
I will usually put new information for the course here on my "private"
INFORMATION about the REDUCED SYLLABUS for this
semester is now here.
A more detailed version of the syllabus is now
To go to the OFFICIAL website for this course click
That site also contains ESSENTIAL information, exercises, text book,
previous examinations, etc.
To go directly to download or read the text book for this course, written
by Allan Pinkus and Samy Zafrany, click here.
Information about your lecturers, and how to contact them is
Here is some VERY IMPORTANT information
about (1) the format of the
tests/exams, (2) homework exercises, (3) the calculation of your tsiyun
sofi, (4) miluim during the semester or test or exam, (5) students with
learning disabilities or other problems. PLEASE BE SURE TO READ THIS. Not
reading it could seriously effect your final grade.
16.4.02 The solution to Moed bet (14.4.02) is here. One version of the exam
18.3.02 Thanks for waiting. Your grades=marks are on the noticeboard on
the 2nd floor of the Amado Building. If you look at the key on the last
page you will see that there is a lot of information there, i.e. results
for individual questions etc. Sorry it took so long. Congratulations to
all who did well, especially those who got 100 or more. (Sorry, your
official grades=marks will be "only" 100.) To those who did not do so
well, best wishes for improved results soon. As you can see, in general,
the results are very good. I take this to mean that most of you took this
course seriously. I wish you success in using this information in your
future studies and work.
14.3.02 Some copies of part 2 of the examination are now also available
from the same envelope containing part 1, next to my door. You can of
course also now get part 2 from this site.
13.3.02 The solution and examination (27.2.02) now include also Question
3 of part 2. (See below.)
4.3.02 The solution to the examination (posted below) has now been
Something had to be fixed in the solution of question 2 in part 1. Thanks
and congratulations to the serious student who first noticed this.
The (revised) and now COMPLETE solution to yesterday's examination is
To remember what the questions were yesterday:
1. You can take a hard copy of part one (yellow page) of the exam from
one of the two
envelopes next to my door. (The other envelope has a different
Please leave those pages for the students who need them.)
2. And you can see a "compact" and now COMPLETE version of the two parts
An example of the picture of 6 graphs which appeared as part of the
examination is available separately here.
22.2.02 Here is a 6 page Hebrew document
which summarizes some of the basic results about Fourier series.
WARNING: A quick summary can be useful, but it is NOT a substitute for
really understanding the material and doing plenty of exercises.
examination this semester you only need the results on the first two pages.
The remaining four pages give the analogous results when the interval
[-pi,pi] is replaced by more general intervals. Thanks to Yoram Yihiye for
helping prepare this document.
This document is NOT a list of formulae to be used during the
examination. The list of formulae which
will appear as part of the examination was posted here on 6/2/02. (See below.)
I have also prepared some other notes. But they are not needed for this
semester's examination. They give a proof of the theorem about the
inverse Fourier transform, using the same ideas as in the alternative
proof of Dirichlet's theorem which was posted here earlier.
The simpler version is here and the general
version is here.
6.2.02 A detailed version of the syllabus is now
6.2.02 (ctd.) Homework exercise No. 4 has been available from the official
site for quite some time I believe. If you have not yet seen it
please go to the official site, and the page of the metargel akhrayey, or
(temporary) link. But note, as we already told you on 16 Feb,
a misprint in
Question 3 part gimel. The term 2tx(t) in the first equation should be
6.2.02 (ctd.) The questionnaire of the examination for Moed Alef, (and
Moed bet,) will include the standard list of formulae that has
appeared on most recent exams in this subject. The exact list that we will
use is here.
It will be provided by US, please do NOT bring your copy. (You will not be
allowed to use it.)
If you are looking for old examinations as a source of questions for
revision, it should be mentioned that some of these are on the page of the
metargel hakhrayi, and some on the main official page of the course.
Many are on both sites, but to get absolutely all of them you currently
have to visit both of these sites. I will request that both of these sites
be updated, to include all examinations, but I do not know when this will
be done. (You can also find links to several examinations and bakhanim
from my page of last semester.)
6.2.02 (ctd.) Probably you know all or most of what you need to know about
partial fractions. (They are sometimes convenient for finding inverse
Laplace transforms.) If you want to know about the completely general case
you can look here.
If two piecewise continous functions f(x) and g(x) both vanish for all
negative x, then their convolution f*g is continuous. Here are the details.
3.3.02 Here is a REVISED VERSION of a
formulation of Fubini's theorem which is useful for changing order of
integration in connection with Fourier transforms. (I noticed that
one condition was missing in the earlier version. The earlier version is
correct if you keep open the option of using Lebesgue integration instead
of Riemann integration.)
21.01.02 Homework exercise no. 3, about Fourier transforms is now
16.1.02 Here is a formulation (Revised now on
2/2/02) of Fubini's
theorem which is useful for changing order of integration in connection
with Fourier transforms.
13.1.02 To find out how to see your bakhan please click
3.1.02 The text of the bakhan of January 1 (in a more compact version)
is here. And
here is a solution.
I plan to rewrite an improved version of the last part of the notes
that I posted here on 26.12.01, i.e. the part about "other intervals". The
formulae there are correct, but there is also a simpler version of them.
29.12.01 Shavua Tov. The second and final part of the solutions/hints
for the homework exercises about inner products is
here. (English, 6 pages. Please look
especially at the first 3 and a half pages. The last two and a half pages
are for "freakim".)
In answer to a question from one student:
The mid term text is without ANY khomer ezer of any kind.
At this stage there are not so many formulae to remember. If you don't
remember them this is perhaps also a sign that you need to do a few more
(But for the final exam we will almost certainly include the "traditional"
two pages of formulae which have been used in the past few years.)
Please remember also that students will not be allowed to leave the
room during the test.
27.12.01 We have decided, after all, to provide solutions for at least
some of the homework problems. Here is
a solution for Question 2 and part of Question 6 from the first set of
homework problems. Maybe there will be additional solutions in the next
26.12.01. Two announcements:
(1) On December 31, 2001 starting from 16:30 until ?? ...
Uri Itai will be available in room 521 Amado to answer questions.
Thank you Uri.
There may perhaps also be other shaot kabala or tirguley hashlama. I
any that I am told about.
(2) In my lecture today I forgot to add a constant to a
formula for integration term by term of Fourier series. (This topic is
NOT for the test on January 1.) For a correction please see here, or read the (now slightly revised)"Term by
Term..." notes mentioned on 18.12.01.
(Those same notes are also
24.12.01 Here is some information about
the bakhan on January 1.
19.12.01 The promised summary about uniform convergence is
18.12.01 You can now get the second set of homework exercises from the
official Fourier series website or from
here. Maybe some of them will not be relevant for the questions of the
midterm test on January 1. We will give exact details later.
In many of the exercises you are asked to determine whether certain
Fourier series converge UNIFORMLY (b'mida-shava) on some set. Uniform
convergence is a topic which you are supposed to remember
from Hedva 1 or Hedva 2. Although in general it is very important, it only
plays a limited role in this particular course.
quick summary, a "Uniform Convergence Survival Kit" containing the main
things you need to know about Unif.Cgce. for this course.
Here is a summary which I wrote last semester (SLIGHTLY REVISED ON
26.12.01) about two topics:
Differentiation and integration of Fourier series, term by term, and
Fourier series on other intervals instead of [-pi,pi].
I plan to rewrite an improved version of the last part about "other
intervals". The formulae there are correct, but there is also a simpler
version of them.
(1) This semester, because of the strike, the last topic "other intervals"
will be discussed in lectures but not
in tirgulim. So we will not ask examination questions about it. (But the
second set of homework exercises includes a "targil reshut" about it.) It
is not a difficult topic to learn yourselves, and you will need it in
other courses, e.g. when you solve differential equations on different
intervals. Yoram Yihiye plans to also provide you with a summary in Hebrew
of the relevant formulae for Fourier series on different intervals.
(2) There is one very small misprint in these notes which I will try to
correct soon. A funny symbol in equation (8) and in some other
places should be a zero.
12.12.01 We will soon be studying the convergence of Fourier series. The
main result is Dirichlet's theorem.
You DO have to be able to use Dirichlet's theorem and state its result
precisely. But this semester, because of the strike, you do NOT have
to know how to prove it.
For those who want to know the proof anyway, or at least a simple
version of it , there are (at least) three options.
(1) You can study the proof in Chapter 2 of the book by Pinkus and
(2) You can read (3 pages, English)
This is a short proof of Dirichlet's theorem in a special case where
the function f is "nice". There is a fourth page with some further
comments and questions.
(3) For a version of the FULL proof which is different from the usual
version in (1) you can read 7 pages (English)
(Part of pages 2 and 3 of this document, is the calculation of the
Fourier series of a particular function, which is an exercise that you
should be able to do, even if you do not care about the proof of
10.12.01 PLEASE SEE BELOW FOR UPDATE ABOUT ROOMS FOR EXTRA LECTURES
10.12.01 Information about the reduced syllabus for this
semester is now here.
As already announced by email,
during the week 9-13 December there will be NO TIRGULIM.
Instead, all four lecturers will give an EXTRA ONE HOUR LECTURE
at the time of the tirgul. So altogether each group will have THREE
HOURS of lectures this week. (The reason of course is to reduce the gap
between lectures and tirgulim caused by the strike.)
Sunday 9 December:
Prof. Michael Cwikel 9:30-10:30 ULMANN 307.
Dr. Yakov Lutsky 14:30-15:30 ULMANN 310.
Wednesday 12 December:
Dr. Alla Shmukler 9:30-10:30 CHURCHILL AUDITORIUM ("sofi").
Dr. Benzion Kon 14:30-15:30 ULMANN 205 ("sofi").
29.11.01 You can find some new (updated) notes (4 pages, English) here.
These notes explain the topics that we are teaching now or will be
teaching soon, (the projection theorem, Bessel's inequality and the
Riemann-Lebesgue lemma) in a slightly different way from the text book.
Now might be a good time for you to read them.
set of homework exercises (about inner product spaces)
here (Hebrew, 3 pages).
But most of
these exercises cannot be solved until you learn a few more topics.
Some of the harder questions are marked with an asterisk(*).
We will wait a few days more to publicize our exact policy about
bakhanim, tsiyun magen etc. because of possible changes to our original
policy because of the strike.
Last semester's page for Fourier S. I. T. is
here. It contains several lists of problems and extra explanations
about material from the lectures, and some examinations and solutions to
some of them. Most of these things can also be useful this semester. As
the semester proceeds I plan to also put more explicit links here to
some of these items.